Have you tried anything suggested so far? Do you have any of the normal restrictions in place ie: user "nobody" allowed to send email, etc? You can add logging to exim and then tail the mainlog file. Also grep “message ID” /var/log/exim_*

Assuming you have root access to WHM. Service Configuration>>Exim Configuration Editor>>Verify the existence of email senders
Also in WHM under Server Setup>>Tweak Settings>>Prevent user nobody from sending email. Also should consider limiting the number of emails per hour allowed to be sent.